Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×#234591 (order 3072) · Inn(G) = D3 · Out(G) = #234591 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C8 |
| direct product | C8×C3⋊C8 |
| unambiguous semidirect product | C3⋊C82 |
| derived, Sylow | C3 ⋊ C82 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | D3 |
| commutator, hypocenter | C3 | C82 |
| Frattini | C42 | C2×D3 |
| Fitting | C3×C4×C8 | C2 |
| socle, prime radical | C22×C3 | C42 |
| cosocle | C3×C42 | C22 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C8 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3)(12,13,14,16,15,17,18,19);(1,2,3)(4,5,6,8,7,9,10,11)96 conjugacy classes — view the full character table.
| maximal subgroup of | C16×C3⋊C8, #67498, #67499, … |
| maximal normal subgroup of | C16×C3⋊C8, #67498, #67499, … |
| maximal quotient of | C8×C3⋊C16, #67437, C16×C3⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 12);